Probability distribution function for reorientations in Maier-Saupe potential

TL;DR

This paper derives an exact analytical probability distribution function for rotational diffusion in Maier-Saupe potentials using confluent Heun functions, applicable to various barrier heights, and calculates related relaxation times.

Contribution

It extends the analytical treatment of the Smoluchowski equation to include Maier-Saupe potentials with variable barriers using confluent Heun functions.

Findings

Exact distribution functions for Maier-Saupe potentials obtained.

Calculated mean first passage times and relaxation eigenvalues.

Results agree with previous approaches across barrier heights.

Abstract

Exact analytic solution for the probability distribution function of the non-inertial rotational diffusion equation, i.e., of the Smoluchowski one, in a symmetric Maier-Saupe uniaxial potential of mean torque is obtained via the confluent Heun's function. Both the ordinary Maier-Saupe potential and the double-well one with variable barrier width are considered. Thus, the present article substantially extends the scope of the potentials amenable to the treatment by reducing Smoluchowski equation to the confluent Heun's one. The solution is uniformly valid for any barrier height. We use it for the calculation of the mean first passage time. Also the higher eigenvalues for the relaxation decay modes in the case of ordinary Maier-Saupe potential are calculated. The results obtained are in full agreement with those of the approach developed by Coffey, Kalmykov, D\'ejardin and their coauthors in the whole range of barrier heights.